Does Blender continue to work in the background despite "No responding", the blue loading circle and the "milky" picture, and is it worth waiting?
Yes indeed it does. No progress bar is because of threading: Blender was expecting the script or whatever to run quickly, or simply can't run multiple threads with the process for whatever reason (EEVEE is an example as Blender cannot multithread OpenGL to EEVEE render and GUI render at the same time), and the GUI updater is waiting for the process to finish.*
*In some rare cases, something could have gone wrong that was also causing the hold-up.